Finite maximal codes and factorizations of cyclic groups
Clelia De Felice
TL;DR
The paper investigates the intricate link between finite maximal codes and factorizations of cyclic groups, building on Zhang and Shum’s structural result and providing a new, simpler proof. It shows that when a finite maximal code $X$ contains $a^n$ with $n$ divisible by at most two primes, the corresponding $X_w$ sets arrange into matrices whose row/column factor sets realize factorizations of $\mathbb{Z}/n\mathbb{Z}$, enabling new conclusions about the triangle conjecture and related inclusion questions. It develops the companion-factorization framework, connects Hajós and Krasner factorizations to good arrangements, and proves decidability results for inclusion problems in low-prime-factor cases, thereby advancing understanding of the factorization conjecture and its weaker forms. The work also establishes partial results for the triangle conjecture and outlines a path to broader decidability and constructive results via Hajós/Krasner structures, with potential impact on algorithmic verification of code properties in algebraic combinatorics of languages.
Abstract
Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture, proposed by Perrin and Schützemberger. The latter concerns finite codes $Y$ which are subsets of $a^* B a^*$, where $a$ is a letter and $B$ is an alphabet not containing $a$. A structural property of finite maximal codes has recently been shown by Zhang and Shum. It exhibits a relationship between finite maximal codes and factorizations of cyclic groups. With the aim of highlighting the links between this result and other older ones on maximal and factorizing codes, we give a simpler and a new proof of this result. As a consequence, we prove that for any finite maximal code $X \subseteq (B \cup \{a \})^*$ containing the word $a^{pq}$, where $p,q$ are prime numbers, $X \cap a^* B a^*$ satisfies the triangle conjecture. Let $n$ be a positive integer that is a product of at most two prime numbers. We also prove that it is decidable whether a finite code $Y \cup a^{n} \subseteq a^* B a^* \cup a^*$ is included in a finite maximal code and that, if this holds, $Y \cup a^{n}$ is included in a code that also satisfies the factorization conjecture.